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Automatically determining knot number and positions is a fundamental andchallenging problem in B-spline approximation. In this paper, the knotplacement is abstracted as a mapping from initial knots to the optimal knots.We innovatively introduce a deep neural network solver to approximate themapping. The neural network is composed of several subnetworks. Each subnetworkis designed to approximate the optimal knot positions in the case of fixed knotnumber. All the subnetworks are stacked together to find the optimal knots(including knot number and knot positions) within some given tolerance. Owingto the powerful approximation capabilities, as well as mature algorithmsdeveloped in deep learning, the proposed method can effectively and efficientlyfind the optimal knot number and knot positions. Several numerical examples aredemonstrated to show the superiority of our method.

A Deep Neural Network for Knot Placement in B-spline Approximation